INF 223, Spring 2015, Plan of Lectures (Status 08.05.2015)

Nr. | Date | Week | Topics | Pages | |

1. | 15.01.14 | 3 | - What is Category Theory? - shift of paradigm - informal discussion of products, dualization, sums | 5 - 8 | |

2. | 20.01.15 | 4 | - graphs and graph homomorphisms: motivation, examples, definition - opposite graphs - discussion of isomorhisms between graphs | 8 - 12 | |

3. | 22.01.15 | 4 | - composition of maps and identity maps - composition of graph homomorphisms and identity graph homomorphisms - associativity and identity law of composition - definition of category | 12 - 14 | |

4. | 27.01.15 | 5 | - categories Set and Graph - a universal definition of isomorphism - composition of isomorphisms is isomorphism - isomorphisms in Set are bijective maps - some finite categories - representation of finite categories by pictorial diagrams | 14 - 16 | |

5. | 29.01.15 | 5 | - other categories with sets as objects: Incl, Inj, Par - subcategory: examples and definition - Nat and Incl as pre-order categories - discussion associations in class diagrams - composition of relations | 17 - 22 | |

6. | 03.02.15 | 6 | - category Rel - association ends and multimaps - category Mult | 23 - 27 | |

7. | 05.02.15 | 6 | - monoids: examples and definition - monoid morphisms: examples and definition - category Mon of monoids - universal property of lists | 27 - 29 | |

8. | 10.02.15 | 7 | - functors: motivation, definition - functors: examples - product categories - opposite category and contravariant functors | 29 - 32 | |

9. | 12.02.15 | 7 | - identity functors and composition of functors - categories of categories: Cat, CAT, SET, GRAPH - pathes: motivation, examples, definition - path graph and evaluation of paths - categorical diagrams: motivation, definition, examples - commutative diagram: definition and examples - path categories | 32 - 33 | |

10. | 17.02.15 | 8 | - summary of the first lectures about "structures" - general discussion about models and metamodels - discussion of a "metamodel" MG of graphs - graphs as interpretations of the graph MG in Set - graph homomorphisms as natural transformations - definition of natural transformations | 35 - 39 | |

11. | 19.02.15 | 8 | - natural transformations: composition and identities - definition of interpretation categories - indexed sets as functor category - arrow categories - category of E-graphs - discussion of arrows between arrows - path equations, satisfaction of path equations - model interpretations - reflexive graphs | 39 - 42 | |

12. | 24.02.15 | 9 | - motivation of "typing" by ER-diagrams and Petri nets - type graph and typed graphs and their morphisms - definition slice category - example typed E-graphs | 43 - 48 | |

13. | 26.02.15 | 9 | - equivalence relations and equivalence classes - quotient sets and natural maps - unique factorization of maps - equivalences as abstraction in mathematics - quotient path categories | 48 - 56 | |

14. | 03.03.15 | 10 | - monomorphisms: definition, examples in Set, Graph, Incl - epimorphisms: definition, examples in Set, Graph, Incl - split mono's and epi's - in Set all epi's are split -> axiom of choice
| 56 - 60 | |

| 05.03.15 | 10 | no lecture (Fagkritisk dag)
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15. | 10.03.15 | 11 | - initial objects: definition, examples in , Incl, Set, , Mult, Graph - terminal objects: definition, examples in , Incl, Set, Mult, Graph
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16. | 12.03.15 | 11 | - sum: definition, examples in , Incl, Set, Graph - product: definition, examples in , Incl, Set, Graph
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17. | 17.03.15 | 12 | - motivation pullbacks: inner join, synchronization, products of typed graphs, pre-images - pullbacks: definition, examples in , Incl, Set, Graph - preimages as pullbacks | ||

18. | 19.03.15 | 12 |
- equalizers, general construction of pullbacks by products and equalizers - equalizers are mono - monics are reflected by pullbacks, coding of monics by pullbacks - composition of pullbacks is a pullback and decomposition of pullbacks | ||

19. | 24.03.15 | 13 | - motivation pushouts: sharing, decomposition of graphs, rule applications - pushouts: definition, examples in , Incl, Set, Graph - coequalizers, general construction of pushouts by sums and coequalizers - discussion: two lines of constructions | ||

20. | 26.03.15 | 13 | - definition of diagrams - cones and limits - co-cones and colimits - completeness and co-completeness - stepwise construction of limits and colimits | ||

| 31.03.15 | 14 | no lecture (Easter Holiday)
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| 02.04.15 | 14 | no lecture (Easter Holiday)
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| 07.04.15 | 15 | no lecture (conference) | ||

| 09.04.15 | 15 | no lecture (conference) | ||

| 14.04.15 | 16 | no lecture (conference) | ||

| 16.04.15 | 16 | no lecture (conference) | ||

21. | 21.04.15 | 17 | - research project: flexible and universal diagrammatic formalism - categorical sketches: example binary relations - criticism of sketch approach - relation = jointly injective/monic - dualization = jointly surjective/epic (cover) | ||

22. | 23.04.15 | 17 | - ER diagrams in DPF - key = injective map - compound attributes = products - formalization of associations: predicate [opp] - diagrammatic signature: definition and examples - atomic constraints - diagrammatic specification | ||

23. | 28.04.15 | 18 | - semantic interpretations in a "semantic universe" U - indexed semantics = interpretation categories - satisfaction of atomic constraints - specification morphisms: definition and example | ||

24. | 30.04.15 | 18 | - revised type graph for ER diagrams - semantics-as-instances - discussion extending type graphs to metamodels - informal discussion of modeling hierarchies | ||

25. | 05.05.15 | 19 | - (revised) specification entailments: examples and definition - graph constraints and universal constraints: examples and validity - discussion: specification entailments give rise to universal constraints | ||

26. | 07.05.15 | 19 | - Universal constraints as transformation rules - model transformation = pushout - discussion: negative application conditions - specification entailments as transformation rules = deduction - example joint formalism and model transformation - discussion: deletion rules | ||

| 12.05.15 | 20 | no lecture (instituttsamling) | ||

| 14.05.15 | 20 | no lecture ( Ascension Day) | ||

| 21 | No more lectures |
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| 22 | No more lectures |
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05.06.15 | 23 | Oral Exam | |||